Three Dimensional Polyhedral Array

Abstract

A polyhedral array comprises a plurality of discrete polyhedrons and a connection network comprising connections that connect the polyhedrons. The discrete polyhedrons are spaced apart from each other at equilibrium in a predetermined generally regular pattern, and each polyhedron is comprised of edges, faces and vertices. The connection network at least partially constrains the discrete polyhedrons relative to each polyhedron's six degrees of freedom. In some embodiments, the connections extend along the bias of the array. In other embodiments, the connections extend along radial directions. All implementations can be constructed as regular arrays or in lattice networks.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a perspective view of a regular icosahedron circumscribed by a cube.

FIG. 1B is a perspective view of a regular dodecahedron circumscribed by a cube.

FIG. 1C is a perspective view of a regular icosahedron illustrating the mutually orthogonal planes that extending between three pairs of opposite edges.

FIG. 1D is a perspective view of a cubooctahedron circumscribed by a cube.

FIG. 1E is a perspective view of a portion of a polyhedral array in which the connections are arranged on the bias.

FIG. 2 is a side elevation view of a portion of the bias direction array shown in FIG. 1E.

FIGS. 3, 4, 5 and 6 are top, bottom, left and right side views, respectively, of an interior polyhedron in the bias direction array of FIG. 1E, showing the positions of the connections to adjacent polyhedrons and their directions.

FIG. 7 is a perspective view of a portion of a radial direction array showing a single interior polyhedron connected to twelve surrounding polyhedrons by radial connections.

FIG. 8 is a perspective view of a portion of a radial direction array showing a single interior polyhedron connected to twelve surrounding polyhedrons by radial connections where the polyhedrons can transform in shape from an ideal shape when subjected to an influence.

FIGS. 9, 10, 11, 12 and 13 are side views of an interior polyhedron of the array of FIG. 8.

FIG. 14A is an exploded perspective view of the interior polyhedron of FIG. 9 and its connections.

FIG. 14B is a side view of an icosahedron element capable of transformation.

FIG. 14C is a side view of an icosahedron element as transformed by a tension force, according to a first example.

FIG. 14D is a side view of an icosahedron element as transformed by a tension force, according to a second example.

FIG. 15 is a perspective view of a bias direction array of a specific implementation where the connections are mechanical elements and the polyhedrons are formed with corresponding connection receiving features.

FIG. 16 is a perspective view of one of the polyhedrons of FIG. 15.

FIG. 17 is a perspective view of one of the connections of FIG. 15.

FIG. 18 is a perspective view of a specific polyhedron construction, showing the polyhedron formed of two geodesic equatorial “saw tooth cut” with a hollow interior.

FIG. 19A is a front view of another array, also referred to as a trigonal lattice.

FIG. 19B is a left side view of the trigonal lattice of FIG. 19A.

FIG. 19C is a top plan view of the trigonal lattice of FIG. 19A.

FIG. 19D is a side elevation view of a repeating element of the trigonal lattice configured with the mechanical elements of FIG. 15.

FIG. 20A is a front view of another array, also referred to as a cubo-octahedral lattice.

FIG. 20B is a left side view of the cubo-octahedral lattice of FIG. 20A.

FIG. 20C is a top plan view of the cubo-octahedral lattice of FIG. 20A.

FIG. 21 is a perspective view of a slice of a polyhedral array absorbing a load and showing some of the polyhedrons moved from their equilibrium positions, with the connections between the polyhedrons omitted for clarity.

FIG. 22 is a perspective view of a slice of a polyhedral array showing one of the polyhedrons rotated from its equilibrium position in response to an applied torque.

FIG. 23 is a perspective view of a polyhedral array implemented as the core of a catalytic converter.

FIG. 24 is a perspective view of a polyhedral array implemented as an artificial reef in a marine setting.

FIG. 25 is a perspective view of a polyhedral array implemented as heat exchange element in a heat exchanger system.

FIG. 26 is a perspective view of a polyhedral array implemented as an artificial biological material.

FIG. 27 is a perspective view of a polyhedral array implemented as a multi-function road and/sidewalk surface.

FIG. 28 is a perspective view of polyhedral arrays implemented as stormwater and erosion control structures.

FIG. 29 is a perspective view of polyhedral arrays implemented as various construction materials.

DETAILED DESCRIPTION

Described below are various embodiments of polyhedral arrays. Each array is comprised of a plurality of discrete polyhedrons, and a connection network comprising individual connections that interconnect the polyhedrons. The connection network serves to constrain each polyhedron with respect to its six degrees of freedom within three-dimensional space, such as is defined by a three-axis Cartesian coordinate reference frame.

Each polyhedron is discrete relative to adjacent polyhedrons in the array. Thus, each polyhedron is separate from the other polyhedrons, and adjacent polyhedrons do not share common vertices, edges or faces. The connections serve to interconnect the polyhedrons, and when the array is in equilibrium, i.e., at rest and not under an applied load, to maintain them in a spaced apart configuration. When the array is subject to an external influence, e.g. a load such as a force or torque applied to the array, one or more of the polyhedrons may be moved in response to the influence. In some implementations, at least one polyhedron, although constrained by its connections, may move from its equilibrium position into contact with an adjacent polyhedron when the array is subjected to a load. Depending upon the specific implementation, the polyhedron's movement may be a translation, a rotation, or a combination of a translation and a rotation. The ability of a polyhedron to move within the array but remain connected to its neighboring polyhedrons allows the array to have “flexibility” under compression and resist the forces being applied.

In general, the array is three-dimensional and may comprise multiple layers depending on the application and the characteristics desired for the application. Each layer is generally planar, and the layers are generally parallel to each other. The connections extend to adjacent polyhedrons in the same layer and in other layers. The connections can be provided in many different configurations and materials to allow the array to have selected properties.

According to one specific implementation, the connections between the polyhedrons extend along the bias (also described as “in bias directions”). In other words, the connection or connections between a first polyhedron and its nearest neighbor are arranged to extend at acute angles relative to the respective plane or plane(s) of these polyhedrons. In one specific implementation, and as described in greater detail below, the direction of the bias or “bias directions” are oriented at about 45 degrees with respect to the plane or planes. At equilibrium, these bias directions are mutually orthogonal, so any connections in the array that do not extend in parallel directions are also orthogonal with respect to each other. Examples of bias direction arrays, which are discussed below in greater detail, are shown in FIG. 2 and FIG. 15.

According to another implementation, the connections between the polyhedrons are described as radial connections. As used herein, “radial connections” for any given polyhedron extend from a face, edge or vertex of the polyhedron in a direction perpendicular to a vector originating its center. Over its length extending towards an adjacent polyhedron, the radial connection changes direction (or “bends”), as is described and illustrated below, such that its opposite ends are oriented in different planes that are mutually orthogonal to each other. Among other benefits, certain implementations of the radial connection array (hereinafter, “radial array”) allow it to deform such that two or adjacent polyhedrons enter into face to face contact when subjected to a load above a predetermined working load range. Examples of radial direction arrays, which are described below in greater detail, are shown in FIGS. 7 and 8.

Polyhedrons and Connections

According to well known geometry principles, one way polyhedrons are described is in terms of the number of faces, vertices and edges that each type of polyhedron has. For example, an icosahedron has twenty faces, twelve vertices and thirty edges. In a regular icosahedron, each of the twenty faces is an equilateral triangle, so all of the angles are equal and the edges are of equal length. In an irregular polyhedron, however, at least some of the angles and edges are unequal. Certain regular polyhedrons and irregular polyhedrons can be used in arrays, as described below and depending upon the specific implementation.

As described above, the array comprises discrete polyhedrons. In specific implementations, a suitable polyhedron is one that, if circumscribed by a cube, has an edge, a face or a vertex approximately coincident with each of the six cube faces. Stated differently, suitable polyhedrons have at least three pair of connection locations, where each pair of connection locations is mutually orthogonal to the others, and each pair of connection features comprises two opposite faces, two opposite edges, two opposite vertices, or two opposite features of different types (e.g., one face and one opposite edge, or one edge and one opposite vertex).

For example, as shown in FIG. 1A, a regular icosahedron 50 can be circumscribed by a cube 52 such that three pairs of the icosahedron's edges are coincident with the faces of the cube 52. Thus, there are (1) a first pair of opposite edges, including a left edge 54 and an opposite right edge (obscured, but along line 56), (2) a second pair of opposite edges, including a top edge 58 and an opposite bottom edge (obscured, but along line 60) and (3) a third pair of opposite edges, including a front edge 62 and an opposite back edge (obscured, but along line 64), each of which is coincident with a respective one of the six faces of the cube 52.

Similarly, as shown in FIG. 1B, a regular dodecahedron 70 can be circumscribed by a cube 72. Thus, three pairs of opposite edges of the dodecahedron (74 and 76, 78 and 80, and 82 and 84) are coincident with the faces of the cube 72. Referring to FIG. 1D, a cubooctahedron 86 can be circumscribed by a cube 88. Thus, the three pairs of opposite square faces of the cubooctahedron 86 (one of each pair being shown at 90, 92 and 94) are coplanar with the faces of the cube 88.

FIG. 1C is another illustration of a regular icosahedron, such as is shown in FIG. 1A, except in FIG. 1C the icosahedron is depicted in a slightly different orientation for better perspective and with transparent faces. As shown in FIG. 1C, each pair of opposite edges of the icosahedron lies in the same plane, and the three planes defined by the three pair of opposite edges are mutually orthogonal to one another.

Referring again to the circumscribing cube, each polyhedron in the array, like an object in space, can be described as having as many as six degrees of freedom, namely the ability to translate in the X, Y and/or Z directions as defined by a fixed coordinate axis (see FIG. 1C), and/or to rotate about one or more of these axes. These rotations are sometimes described as “pitch,” “yaw” and “roll.” The connections serve to constrain some or all of each polyhedron's six degrees of freedom in at least a limited way. In some specific implementations, the array can be configured such that the connections among the polyhedrons eliminate one or more degrees of freedom for at least one polyhedron (i.e., constrain the polyhedron to substantially no movement within that degree of freedom).

Of the five regular Platonic polyhedrons, namely the tetrahedron (4 equilateral triangle faces, 6 edges and 4 vertices), the cube (6 square faces, 12 edges and 8 vertices), the octahedron (8 equilateral triangle faces, 12 edges and 6 vertices), the dodecahedron (12 regular pentagon faces, 30 edges and 20 vertices), and the icosahedron (20 equilateral triangle faces, 30 edges and 12 vertices), only the regular tetrahedron and the regular octahedron do not provide at least three pair of connection locations that are mutually orthogonal to one another (although a special case of the octahedron does meet this criterion). Many other polyhedrons other than the Platonic polyhedrons are also suitable, such as truncated versions of the cube, octahedron and icosahedron (also referred to as a “fullerene” or a “bucky ball”), cubo-octahedrons, dodecahedrons, icosidodecahedrons, rombicosidodecahedrons, snub dodecahedrons, etc. Stellated forms, deltahedrals, dual tetrahedrals and other similar variations of Platonic and Archimedian polyhedrons are also suitable. The specific polyhedrons enumerated here are not to be construed as an exhaustive list, as other polyhedrons having the appropriate number and arrangement of connection features are also suitable

Two important polyhedrons are the icosahedron and the truncated icosahedron. As described above, the icosahedron is a practical choice because it has three pairs of parallel faces. The truncated icosahedron is derived by “slicing off” the twelve vertices of the icosahedron, thereby forming twelve regular pentagons. The truncated icosahedron has twelve regular pentagonal faces, twenty regular hexagonal faces (for a total of 32 faces), 60 vertices and ninety edges. The pentagons and hexagons provide a plurality of pairs of parallel surfaces.

Polyhedrons with fewer rectangular faces (including square faces) are generally more desirable because the triangular faces are more stable than rectangular faces. Although the cube is an acceptable polyhedron with which an array can be formed, in practice, other polyhedrons offer greater advantages.

In some implementations, it is possible to form a mixed array that comprises more than one type of polyhedron. For example, it is possible to form a mixed array having icosahedrons and truncated icosahedrons. Other such mixed arrays are also possible. It is also possible to use polyhedrons of different sizes in the same array, particularly if the differently sized polyhedrons are geometrically scaled relative to each other.

The polyhedrons in the array may be solid, or they may have solid faces that define a hollow interior. One or more faces of a polyhedron may have openings defined therein. It is also possible in some implementations to use polyhedrons having a “wireform” configuration with material defining only the edges and the faces being open. In some implementations, the polyhedrons include enhancements departing from ideal regular forms to facilitate connecting the polyhedrons to each other, such as is described in connection with one implementation shown in FIGS. 15 and 16.

Connections as used herein refers to any matter, including any arrangement of matter (such as a structure), or material that serves as part of the connection network in connecting the polyhedrons together and maintaining their spaced apart configuration in the array at equilibrium. Connections can include mechanical elements, molecules or bonds, such as guest molecules, ligands, ligatures, etc.

Each connection can be a separate element, or multiple connections can be formed together, depending upon the particular geometry, materials and requirements of the array. In arrays with one than one type of connection locations, e.g., vertices and faces, there may be connections extending between a vertex at one end and a face on the opposite end.

Response to Loading

As described above, the polyhedral arrays can be designed to withstand loadings within a predetermined working range. For example, the polyhedral arrays can be designed to withstand forces and torques, including compression forces, tensile forces and torsional forces applied to the array, as well as other forces, such as shear forces, that may be developed internally within the array. Bending moments and axial compression forces can also comprise part of the loadings. As described, depending upon the magnitude of the loading, one or more of the polyhedrons in the array may be caused to move from its position at equilibrium. FIG. 21 is a perspective view of a slice of a representative polyhedral array P withstanding a substantial force F applied to the array, and most directly, to the polyhedron G. In FIG. 21, the connections are omitted for clarity and to emphasize that the polyhedron G has been moved by the force F. More specifically, the force F has urged the polyhedron G to translate (in a generally downward direction). Although the connections are not shown, they serve to assist in absorbing and spreading the force F to other polyhedrons and connections in the array. If the force F is within the expected working range, and if the array P is so designed, the array P (and specifically, the polyhedron G) will return to its original equilibrium state if the force F is removed.

In addition to withstanding forces, the polyhedral arrays also withstand torques. FIG. 22 is a perspective view of a slice of an array Q showing a torque T applied to the array and, most directly, to the polyhedron H. Although not shown, the connections between the H polyhedron H and its neighboring polyhedrons help in dissipating and resisting the torque T, thus allowing the array Q to withstand it. As shown, the torque T is sufficient to move the polyhedron H, i.e., to rotate it approximately 60 degrees, but the array Q remains intact.

The examples of FIGS. 21 and 22 are illustrative of ideal forces and ideal torques. In practice, nearly every loading of an array would include both forces and torques. Also, the examples describe loadings applied to boundaries of the array, such as a surface, edge or corner of the array. In practice, other loadings may also be experienced.

Bias Direction Array

As described above, one embodiment includes connections oriented on the bias. Relative to a face of the array to which a load is applied (or resolved to apply) in a normal direction, the bias directions are defined to mean directions intersecting the face of the array at an angle. In specific implementations, the bias directions can form angles of about 30 to about 60 degrees with the face of the array. Even better attributes are recognized with bias directions of about 40 to about 50 degrees.

Typically, the bias directions extend at about 45 degrees relative to the face of the array, but can be adjusted in orientation to the applied load. Thus, the bias direction array is in direct contrast with an orthogonal array in which the connections extend at approximately 90 degrees or zero degrees relative to the face of the array. Compared to other orientations, connections extending along bias directions provide robust resistance to shear.

FIG. 1E is a perspective view of a bias direction array 10. The array 10 is comprised of multiple polyhedrons 12 arranged in a generally repeating pattern, with connecting elements 14, 16, 18 interconnecting the polyhedrons 12 and extending along bias directions.

In the array 10, which is shown at equilibrium, the polyhedrons 12 are spaced apart and do not contact each other. In FIG. 1E, the array 10 is shown as having three layers for convenience of illustration, with some parts of highly obscured layers omitted for clarity, but as described elsewhere, it could be extended in any direction indefinitely.

The polyhedrons 12 in the array 10 have the same general orientation. In the example of FIG. 1E, the polyhedrons 12 are regular icosahedrons. As shown, the icosahedrons in the bias direction array 10 have a “face up” orientation, i.e., one of the triangular faces is oriented vertically upward.

In the array 10, there is a lower layer 20 of polyhedrons generally occupying a first plane, an intermediate layer 22 above the lower layer 20 and generally occupying a second plane and an upper layer 24 above the intermediate layer 22 and generally occupying a third plane. As best seen from the right side of FIG. 1, the right front face of the array is slanted, i.e., biased, from the protruding lower layer 20 in a rearward direction to the recessed upper layer 24.

Each of the polyhedrons 12 in the lower layer 20 and in the upper layer 24 is an edge polyhedron, i.e. a polyhedron occupying an edge position that defines an edge of the array 10. For the intermediate layer 22, the visible edge 12 are along the front right face (five in number) and along the front left face (two in number). Each edge polyhedron has fewer connections to adjacent polyhedrons than an interior polyhedron.

Although not clearly visible in FIG. 1E, there are interior polyhedrons in the intermediate layer 22 rearward of the visible edge polyhedrons. Each interior polyhedron has a connection to each of twelve surrounding polyhedrons, including the six surrounding polyhedrons in the same layer, three polyhedrons in the immediately adjacent upper layer and three polyhedrons in the immediately adjacent lower layer, as depicted in the illustrated orientation.

The connections 14, 16, 18 are shown in FIGS. 1E and 2 schematically as lines solely for purposes of clear illustration. Of course, each connection can take the form of any suitable connection as is described elsewhere in this application. Referring to FIG. 2, which is side view of a portion of the array 10 shown in FIG. 1E, the connections visible for the edge polyhedrons are shown. Each edge polyhedron has at least three connections to adjacent polyhedrons if it is part of two edges, five connections if it is part of one edge and six connections if it is part of a face.

In FIG. 2, the connections 14 connect adjacent polyhedrons in the plane of the face that are within the same layer (as illustrated in the same horizontal row). The connections 16 extend into the page at an angle and connect polyhedrons within one layer to polyhedrons in a layer above or a layer below. The connections 18 also interconnect polyhedrons in different layers. The mutually orthogonal directions of the connections 14, 16, and 18 is most clearly seen on the coordinate axes shown at the right of FIG. 2.

For the specific implementation of FIGS. 1E and 2, the connections 14, 16, 18 interconnect the polyhedrons 12 at connection locations positioned on edges of the polyhedrons. FIGS. 3, 4, 5 and 6 are side views of an interior polyhedron 12 of the array 10 showing the connections to the twelve surrounding polyhedrons in dashed lines. Considering the “pentagon” sides of FIGS. 3 and 4 to be top and bottom sides, six of the twelve connections can be seen from either of these sides. The remaining sides, called the “front” and “back” sides for convenience, each show four connections.

The polyhedrons 12 in the example of FIG. 1 are regular icosahedrons, but any other suitable polyhedron meeting the criteria described elsewhere in this application could substituted in place of a regular icosahedron, either at all locations or at fewer than all locations

Radial Array

FIG. 7 is a perspective view of a portion of a radial array 110. As described above, the radial connections 114 between polyhedrons 112 extend generally uniformly at right angles relative to the respective polyhedron's center.

As shown in FIG. 7, there are twelve edge polyhedrons surrounding the single interior polyhedron 113. In this example, there are three edge polyhedrons in a lower layer, six polyhedrons in the intermediate layer (three edge polyhedrons and the interior polyhedron being visible), and three edge polyhedrons in the upper layer. Overall, the twelve edge polyhedrons occupy the same positions as the vertices of a cubo octahedron.

As shown in FIG. 7, the connections 114 in the radial array connect different planes. Stated differently, each connection has a first end connected at a first plane on a first of the polyhedrons, and a second end connected at a second plane extending and translating orthogonally onto a second of the polyhedrons. In the specific implementation shown in FIG. 7, the connections 114 terminate at faces of the polyhedrons. Thus, each connection does not extend between closest faces of adjacent polyhedrons, but instead can be said to “bend” to connect a next-closest face.

In the radial array, the polyhedrons in the different layers as shown in FIG. 7 are oriented in face to face contact. Thus, in response to an influence, one or more aligned polyhedrons may be forced together and into face-to-face contact. Polyhedrons in face-to-face contact provide a strong and stable configuration (similar to a column) for resisting compressive forces applied to the array. Also, the radial array is particularly resistant to torsional loads applied to the array, even a torque applied to a single peripheral polyhedron, as the loaded polyhedron's connections to adjacent polyhedrons help withstand and spread the torque.

FIG. 8 is a perspective view of another implementation of a radial array 210, shown from a different perspective than FIG. 7, in which the polyhedrons can transform in shape in response to an influence. For example, one or more of the polyhedrons 212 can transform in shape, i.e. morph or skew, from its equilibrium shape as shown in FIG. 14B without breaking. FIG. 14C shows one of the polyhedrons 212 after it has been transformed from its equilibrium configuration by a generally single-axis tension force C applied to two opposite faces of the icosahedron. The resulting transformed icosahedron 212′ has been elongated in the directions of the force C. In FIG. 14D, the resulting transformed polyhedron 212″ is the result of a tension force D acting on the polyhedron, which causes it to elongate in the directions of that force as well as in directions E perpendicular to the force D. The same transformed polyhedron 212″ results if the force is applied in the same plane as the force D, except rotated 90 degrees (i.e., if the force is applied along a direction normal to the page).

The polyhedrons 212 can be provided with predetermined features to facilitate the transformations in shape. In the specific implementation of FIG. 8, the polyhedrons 212 are icosahedrons formed of multiple pieces. Twelve of the faces of the icosahedron have a connection element, such as a protruding post or hub 222, for connecting the polyhedron to other polyhedrons. The remaining eight faces of the icosahedron are formed of three overlapping portions arranged to slide or “shutter” over each other, thus allowing the icosahedron to transform in shape. In addition, the connections 214 have ends that are rotatably connected to respective faces of the polyhedrons to which they are connected, thus allowing the connections 214 to pivot relative to the polyhedrons. For the connection 216, it can be seen that the ends 218, 220 are pivotably connected to respective hubs 222, 224, as one example.

FIGS. 9, 10, 11, 12 and 13 are side views of an interior polyhedron 212 with the connections 214 and their general orientation shown (but in a reduced length for clarity of illustration).

FIG. 14A is an exploded view the polyhedron 212 shown with twelve connections 214, i.e., suitable for positioning it as an interior polyhedron of an array. The polyhedron 212 is constructed of 12 interconnecting components 230 that together form a generally regular icosahedron with 20 faces with overlapping portions. As shown, each of the twelve components has one of the twelve hubs 222 of the icosahedron, several of which are visible in the drawing. The assembled components 230 are movable relative to each other to allow the icosahedron to transform in shape.

Attributes of Arrays

The polyhedral arrays described herein are “omni-extensible,” which is to say that an existing array can be extended in any direction by joining additional polyhedrons with additional connections. It is not necessary to first disassemble the existing array before extending it. In the same way that an existing array can be extended, it also possible to join two or more arrays into a single larger array.

When the polyhedral arrays are configured in a regular pattern, the equilibrium locations of the polyhedrons in the array are unique and thus can be specified, e.g., using a Cartesian coordinates or other similar system. Thus, the position of each polyhedron in the array can be described as being “addressable.” In some implementations, at least some portion of the array is capable of carrying or holding electric charge, and the polyhedrons or connections in this portion of the array can be individually addressed to store data and/or to convey signals or transmit power.

As shown in the examples, the polyhedral arrays at equilibrium have spaces separating the polyhedrons. As also described, additional spaces at regularly repeating intervals can also be designed into the array. Depending upon the scale at which the array is constructed, as well as the selected length(s) of the connections relative to the size of the polyhedrons, the resulting array can be designed to provide a desired permeability suitable for a particular application. Thus, an array configured as a building material to be installed in a generally horizontal orientation in an exposed environment can be configured for permeability to rain and storm water, yet still provide the structural strength to support the expected loads for the anticipated design life.

As shown in some of the application examples described below, the configuration of the polyhedral arrays allows for them to interface with other structures, such as, e.g., conventional building materials. Many common conventional building materials have a generally rectangular prism shape, such as conventional bricks, dimensional lumber, plywood and other types of sheet material, etc., which is defined by 90 degree angles. Although most examples of the polyhedral arrays do not terminate at edges defining 90 degree angles, the same mutually orthogonal connection locations of the polyhedrons, some of which are free for edge polyhedrons, can be used for dimensionally reliable and stable attachment to other adjacent materials and structures.

In the described examples, the polyhedral arrays are shown as comprising polyhedrons of the same general size, which is a typical configuration. In some arrays, however, it is possible to mix polyhedrons of different types, or to substitute polyhedrons of different sizes. For example, an array can be formed where the majority of polyhedrons are regular icosahedrons having a unit size, and a larger icosahedron having a size that is a geometric multiple of the unit size is substituted into the array periodically.

A polyhedral array can be configured to be isotropic, i.e., to have the same properties in all directions. Alternatively, the array can have a predetermined anisotropy, e.g., to address a particular requirement of the specific implementation. Also, the polyhedrons and the connections can be isotropic or anisotropic.

The arrays can be configured to have different portions exhibiting different properties. For example, the polyhedrons occupying the edge positions in an array could be formed of a material more resistant to environmental conditions, or specifically adapted for receiving a thin covering layer or attaching to a conventional adjacent structure. Polyhedrons having different physical properties, including elasticity, density, melting point, strength in compression, etc., to name a few, could be substituted in the array to achieve a desired result. Moreover, connections can be adapted in the same way. The properties of the connections can be varied according to their location in the array, their orientation relative to the expected load, etc. In addition, connections can be designed to exhibit different properties along their length.

As an example, the relative rigidities of the polyhedral elements and the connections elements can be tailored for a given application. In addition, individual instances of the same type of element, e.g., the connections, may be provided with varying rigidities to provide a desired anisotropy to the array. For example, connections extending in the “z” direction may be made less or more rigid than the interconnecting members in the “x” and “y” directions where the anticipated loading configuration differs in the “z” direction as compared to the “x” and “y” directions.

In some cases, the polyhedral arrays deviate from perfect regularity yet still have the overall function and behavior of a regular array. In some cases, a polyhedron or a connection may be missing, or there may be a foreign object or impurity present in the array instead of one of the polyhedrons. Indeed, certain implementations warrant slight departures from perfect regularity to account for specific local conditions.

In general, the polyhedrons and connections may be made of any material suitable for the particular application. For large scale applications, familiar materials such as metals, alloys, composites, plastics and others may be used.

It is also possible to provide very small scale arrays, such as at the nanoscale or microscale. Some molecular forms have specific polyhedral geometry. As two examples, the C₆₀ molecule is a truncated icosahedron, and the B₁₂ molecule is an icosahedron. Work in the area of these and similar molecules is represented by U.S. Pat. No. 6,531,107 entitled “Fabrication of Molecular Nanosystems,” U.S. Pat. No. 6,841,456 and U.S. Pat. No. 6,965,026 entitled “Nanoscale Faceted Polyhedra” (these references are incorporated herein by reference). Assembly techniques, such as atomic force microscopy or self assembly, may be used to provide suitable molecules as polyhedral elements. Further, and not appreciated until now, the polyhedral molecules can be arranged into predetermined arrays, such as bias direction arrays or radial arrays, with connections designed as described above and formed of molecules, ligands or ligatures to give the resulting arrays overall properties useful in the design and fabrication of larger arrays (such as lattices) and objects.

Specific Implementations

According to one specific implementation as shown in FIGS. 15, 16 and 17, polyhedral arrays can be constructed using mechanical elements, such as icosahedral elements 250 and connector elements 252.

FIG. 15 is a perspective view of a portion or “chunk” of a bias direction array 254. As shown, the icosahedral elements 250 and the connector elements 252 are arranged generally as shown for the bias direction array of FIG. 2, i.e., with the connector elements 252 extending on the bias relative to the planes defined by the layers of icosahedral elements 250.

Referring to FIG. 16, each icosahedral element 250 has twelve connection locations 256 positioned about the outer periphery of the icosahedral element, eight of which are visible in FIG. 16. The twelve connection locations 256 are arranged in six pairs, i.e., three sets of oppositely arranged pairs, and each of these three sets has an orientation mutually orthogonal to the other two sets.

In the illustrated embodiment, each of the connection locations 256 is positioned within an opening in a projection 258 protruding slightly from the icosahedral element's outer surface and extending across one of its edges.

As best shown in FIG. 17, each connector element 252 has a first end 260, an opposite second end 262 and a body 264 between the ends 260, 262. Each end 260, 262 is shaped to connect with one of the connection locations 256 by insertion of the end into one of the openings defined in the projections 258. Once inserted, the end 260 or 262 terminates near the edge of the icosahedral element 250 (which is covered by the projection 258).

As also shown in FIG. 17, each end 260, 262 can be tapered to provide a proper fit within the opening and orientation relative to the polyhedron. Also, each end 260, 262 can have retaining elements 268 extending alongside and positioned to project into edge apertures 266 provided in the icosahedral element 250 to assist in retaining the connector elements 252 in place.

The body 264 of the connector can have a spring-shaped construction as shown. Other geometries are, of course, also possible. The spring-shaped body allows the connector to constrain the icosahedral elements 250 to which it is attached, yet can resiliently deform when those icosahedral elements are subjected to a force or a torque.

In the specific embodiment, the icosahedral elements 250 are made of a plastic and have a hollow construction. The connector elements 252 are also made of a plastic.

According to some implementations, providing polyhedrons of a hollow construction can be achieved by providing polyhedron halves having specific geometries. Referring to a specific implementation for the icosahedron shown in FIG. 18, each icosahedral element half 270 can be formed such that when the halves are mated together, the junction follows a geodesic saw tooth-shaped equator 272.

FIG. 19A is a side elevation view of another implementation of an array 280, also referred to as a lattice. In the array 280, there is at least one layer in the array that includes open spaces of a predetermined size at and location at equilibrium. These open spaces are defined by regions that are large enough to accommodate at least one additional polyhedron, but are vacant. Arrays with predetermined open spaces offer advantages, such as lighter weight per unit area or volume than corresponding filled arrays (including, e.g., the bias direction array of FIG. 1).

FIGS. 19B and 19C are left side elevation and top plan views, respectively, of the array 280. For clarity of illustration, the connections between the individual polyhedrons have been omitted.

In the array 280, which is also referred to herein as a “trigonal” lattice, the repeating unit 282 is seven polyhedrons, including three connected polyhedrons 284 in the lower layer, three connected polyhedrons 286 in the upper layer, and a single polyhedron 288 in the intermediate layer connected to each of the polyhedrons 284, 286 in the lower and upper layers. From FIG. 19C, it can be seen that there are eight repeating units 282 shown.

In the drawings, the spacing between adjacent repeating units 282 has been exaggerated slightly for clarity. In practice, the space between polyhedrons in the lower and upper layers that are adjacent to each other, but of different repeating units, can be the same as the space separating adjacent polyhedrons within the same repeating unit.

The single polyhedron 288 in the intermediate layer is not connected to other intermediate layer polyhedrons. Thus, there is no connection between the intermediate layer polyhedron 288 and an adjacent intermediate layer polyhedron, such as the intermediate layer polyhedron 290. Within the intermediate layer, predetermined recesses or spaces S are defined between the polyhedrons in that layer (and are bounded by the polyhedrons in the lower and upper layers). The spaces S generally occur according to a periodic pattern. The size and frequency of spaces within the array can be selectively determined according to properties desired for the specific application of the array.

FIG. 19D is a side elevation view of a repeating unit 282′ for a trigonal lattice, which is similar to the repeating unit 282 in geometry, but in this specific implementation is formed of mechanical polyhedron elements 250 and connector elements 252.

As another example, FIG. 20A is a front elevation view of an array 292 according to another implementation. In the array 292, there is a three-layer repeating unit 294 of twelve polyhedrons, including a central polyhedron surrounded by and connected to six polyhedrons in the intermediate layer, and connected to three polyhedrons in the lower level and three polyhedrons in the upper layer.

FIGS. 20B and 20C are left side elevation and top plan views, respectively, of the array 292. For clarity of illustration, the connections between the individual polyhedrons have been omitted.

As shown, the array 292 as illustrated includes two layers of repeating units 294 with six repeating units per layer, for a total of twelve repeating units. Referring to the figures, assuming the exterior polyhedrons of the repeating unit 294 are vertices, the repeating unit is a cubo-octahedron. Thus, the array 292 can be described as a “cubo-octahedron” lattice.

Referring to the figures, each repeating unit 294 is connected to any adjacent repeating unit(s) in the same horizontal “row” or vertical “column.” Among the interconnected repeating units 294 that comprise the array, however, there are predefined spaces T that occur at each intersection of eight repeating units 294. In addition to the full spaces T, there are additional smaller spaces U between at each intersection of four repeating units.

Applications

FIG. 23 shows a polyhedral array implemented as a core or substrate 402 of a catalytic converter 404 for a vehicle, e.g., an automobile. The individual polyhedrons can be spaced apart as necessary to provide the proper air flow through the catalytic converter. Also, the properties of the polyhedron provide for substantial surface area per volume, which increases the area for chemical reactions to occur.

FIG. 24 shows a polyhedral array implemented as an artificial reef 406 in a marine setting. In the artificial reef 406, the polyhedrons have open faces and hollow interiors that permit the reef to sink and allow water and organisms to circulate through the structure. Also, the polyhedrons can be configured to enclose guest nutrients that become available to organisms in the water as through an active delivery system or a passive erosion of the artificial reef.

FIG. 25 shows a heat exchanger, such as an air conditioning unit, with the heat exchange elements 410 implemented as polyhedral arrays. The heat exchange elements 410 can be sized as desired to provide the desired flow rates and mixing in the air flow passages and in the heat exchange passages, as well as to achieve other performance goals.

FIG. 26 shows a polyhedral array implemented as an artificial biological material, in this case an artificial human bicep muscle portion 412. The muscle portion 412 can be configured in accordance with predetermined requirements, e.g., tensile strength, flexibility, elasticity, desired change in shape through the associated limb's range of motion, etc.

FIG. 27 shows a polyhedral array implemented as a multi-layer construction material 418. The material 418 can be configured to have a road surface portion 420 capable of withstanding vehicle weights. The road surface portion 420 can be porous to provide drainage, but if covered, an integrated drainage portion 422 can be included to drain away storm water, etc. from the road surface portion 420. On the adjacent sidewalk, the material 418 can have a plant support portion 424. Optionally, the array of the plant support portion can be configured for integrated irrigation of plants. At a curb portion 426 separating the road surface portion 420 from an adjacent sidewalk portion 428, the array can be configured for integrated charge and/or signal carrying capability to provide illumination (e.g., warning lights, street lights, traffic signal lights) and/or information (warnings, street names, etc.). Similar to the road surface portion 420, the array of the sidewalk portion 428 can be left exposed to provide a porous construction or it can serve as a base for receiving a solid outer layer.

FIG. 28 shows a polyhedral array implemented as storm water and erosion control structures, such as a multi-level culvert or canal wall 430 construction. Also shown is a polyhedral array implemented with earthen layers as part of a reinforced flood wall 432 adjacent one side of the canal wall 430. On the other side of the canal, another polyhedral array implemented as part of a vehicle and/or pedestrian path 434, together with other earthen layers.

FIG. 29 shows a house 435 under construction with polyhedral arrays implemented as exterior building panels 436, interior high-strength safe room panels 438, interior panels 440 and roofing material 442, to name just a few. All of the polyhedral arrays as implemented are configured for compatibility with conventional dimensional lumber framing and other conventional building materials.

In addition to bias direction arrays and radial arrays, other array configurations are of course also possible, provided that the polyhedrons are discrete from each other at equilibrium and the connections interconnect the polyhedrons and constrain their movement to a desired degree in the resulting array.

The terms and expressions that have been employed in the foregoing specification are used as terms of description and not of limitation, and are not intended to exclude equivalents of the features shown and described or portions of them. In view of the many possible embodiments to which the disclosed principles may be applied, it should be recognized that the illustrated embodiments are only preferred examples and should not be taken as limiting in scope. Rather, the scope is defined by the following claims. I therefore claim all that comes within the scope and spirit of these claims.

Claims

1. A polyhedral array, comprising: a plurality of discrete polyhedrons that are spaced apart from each other at equilibrium in a predetermined generally regular pattern, each polyhedron being comprised of edges, faces and vertices; anda connection network comprising connections extending along bias directions to connect the polyhedrons;wherein the connection network at least partially constrains the discrete polyhedrons with respect to each polyhedron's six degrees of freedom.

2. The array of claim 1, wherein each polyhedron's six degrees of freedom are defined as the ability to translate in the X, Y and/or Z directions of a coordinate reference frame, and the ability to rotate about the X, Y and/or Z directions.

3. The array of claim 1, wherein the polyhedrons are arranged in multiple, generally parallel layers.

4. The array of claim 1, wherein at least some of the plurality of polyhedrons generally occupy a first plane, and wherein the bias directions along which the connections extend intersect the first plane.

5. The array of claim 3, wherein the bias directions are oriented at angles of approximately 45 degrees to the first plane.

6. The array of claim 1, wherein the polyhedrons are arranged in multiple layers, and wherein the bias directions along which the connections extend are inclined at about 45 degrees relative to an expected direction of a resolved load on the array.

7. The array of claim 1, wherein at least one of the connections extends between an edge of a first of the polyhedrons and an edge of a second of the polyhedrons.

8. The array of claim 1, wherein at least one of the connections extends between a face of a first of the polyhedrons and a face of a second of the polyhedrons.

9. The array of claim 1, wherein at least one of the connections extends between a vertex of a first of the polyhedrons and a vertex of a second of the polyhedrons.

10. The array of claim 1, wherein at least one of the connections extends between one of a group consisting of a face, an edge and a vertex of a first polyhedron, and a different one of the group consisting of a face, an edge and a vertex of a second polyhedron.

11. The array of claim 1, wherein the array is coherent.

12. The array of claim 1, wherein the array is omni-extensible.

13. The array of claim 1, wherein an existing array can be increased in size by connecting additional discrete polyhedrons to existing discrete polyhedrons with additional connections without other modifications to the existing array.

14. The array of claim 1, wherein each discrete polyhedron is a finitely closed structure having structural integrity independent of the respective connections to which said discrete polyhedron is connected and independent of other discrete polyhedrons in the array.

15. The array of claim 1, wherein the interconnecting network is configured in a generally repeating pattern.

16. The array of claim 1, wherein the connections comprise at least one of mechanical elements, bonds or guest molecules, ligands and ligatures.

17. The array of claim 1, wherein the connections comprise mechanical elements that have greater resiliency than the polyhedrons.

18. The array of claim 1, wherein the connections are mechanical elements comprising spring-shaped portions.

19. The array of claim 1, wherein each polyhedron occupies a unique location at equilibrium that can be specified with Cartesian coordinates.

20. The array of claim 1, wherein the array is anisotropic, with at least one polyhedron having specific properties different from another of the plurality of polyhedrons.

21. The array of claim 1, wherein at least one of the connections can have predetermined properties different from another of the connections.

22. The array of claim 1, wherein at least one of the connections has a property that varies along its length.

23. The array of claim 1, wherein the discrete polyhedrons comprise a first material and the connections comprise a second material different from the first material.

24. The array of claim 1, wherein at least one of the plurality of polyhedrons comprises a closed polyhedron having a majority of closed faces.

25. The array of claim 1, wherein at least one of the plurality of polyhedrons comprises an open polyhedron each having a majority of open faces.

26. The array of claim 1, wherein the plurality of polyhedrons comprises at least one interiorly hollow polyhedron.

27. The array of claim 1, wherein the plurality of polyhedrons comprises at least one solid polyhedron.

28. The array of claim 1, wherein each polyhedron satisfies the condition of having a face, an edge or a vertex approximately coincident with one of the six sides of a cube circumscribing the polyhedron.

29. The array of claim 1, wherein the polyhedrons are selected from the list consisting of cubes, icosahedrons, truncated cubes, truncated octahedrons, truncated icosahedrons, cubo-octahedrons, dodecahedrons, truncated dodecahedrons, icosidodecahedrons, rombicosidodecahedrons, snub dodecahedrons, truncated cubo-octahedrons, stellated forms, deltahedrals and dual tetrahedrals.

30. The array of claim 1, wherein at least one of the polyhedrons is formed of two halves.

31. The array of claim 30, wherein the polyhedron is an icosahedron and has a geodesic saw tooth-shaped equator defining the two halves.

32. The array of claim 1, wherein at least one of the connections extends continuously from a first polyhedron, to a second polyhedron and to an nth polyhedron.

33. The array of claim 1, wherein at least some of the connections are compression members configured primarily to resist compression forces.

34. The array of claim 1, wherein the predetermined regular pattern in which the polyhedrons are arranged can include spaces occurring at generally regular uniform intervals.

35. The array of claim 1, wherein at least some of the polyhedrons are icosahedrons.

36. The array of claim 1, wherein the plurality of polyhedrons includes at least one interior polyhedron connected to twelve adjacent polyhedrons.

37. The array of claim 1, wherein the discrete polyhedrons can move relative to each other in response to applied loads, but remain spaced apart from each other at equilibrium and within a selected working load range.

38. The array of claim 1, wherein a force above a predetermined working load range applied to the array can urge two adjacent polyhedrons from an equilibrium position in which the adjacent polyhedrons are separated from each other into an under load position in which the adjacent polyhedrons are in contact with each other.

39. A polyhedral array, comprising: a plurality of discrete polyhedrons that are spaced apart from each other at equilibrium in a predetermined generally regular pattern, each polyhedron being comprised of edges, faces and vertices; anda connection network interconnecting the discrete polyhedrons, the connection network comprising radial connections, wherein a first end of each radial connection and a corresponding first connection location on one of the polyhedrons occupy a first plane, and wherein a second end of each radial connection and a corresponding second connection location on another of the polyhedrons occupy a second plane that is not parallel to the first plane,wherein the interconnecting network at least partially constrains the discrete polyhedrons relative to each polyhedron's six degrees of freedom.

40. The array of claim 39, wherein the first and second planes of the respective first and second connection locations are mutually orthogonal.

41. A polyhedral array, comprising: a plurality of discrete polyhedrons that are spaced apart from each other at equilibrium in a generally regular pattern, each polyhedron being comprised of edges, faces and vertices; anda connection network interconnecting the discrete polyhedrons, the connection network comprising connections having a first end connected in a first plane and a second end connected in a second plane different from the first plane,wherein the connection network at least partially constrains the discrete polyhedrons relative to three Cartesian axes, and adjacent polyhedrons are oriented in a face to face relationship relative to each other.

42. The array of claim 41, wherein the connections are curved mechanical elements, and wherein each curved mechanical element has a first end connected to a first face of a first polyhedron and a second end connected to a second face of a second polyhedron, the first and second faces occupying different planes.

43. The array of claim 41, wherein when the array is subjected to a load above a predetermined working load range, the array deforms such that at least two of the polyhedrons make face to face contact with each other.

44. The array of claim 41, wherein when the array is subjected to a force or torque above a predetermined working range, the array deforms such that at least one of the polyhedrons undergoes a predetermined transformation in shape.

45. The array of claim 44, wherein the at least one of the polyhedrons is reversibly transformed such that the at least one of the polyhedrons returns to an original shape if the force or torque is removed.